Precise Rates in the Generalized Law of the Iterated Logarithm in ${\mathbb{R}}^m$
Received:February 06, 2017  Revised:August 04, 2017
Key Word: precise rates   law of iterated logarithm   complete convergence   i.i.d. random vectors
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.61662037) and the Scientific Program of Department of Education of Jiangxi Province (Grant Nos.GJJ150894; GJJ150905).
 Author Name Affiliation Mingzhou XU School of Information and Engineering, Jingdezhen Ceramic University, Jiangxi 333403, P. R. China Yunzheng DING School of Information and Engineering, Jingdezhen Ceramic University, Jiangxi 333403, P. R. China Yongzheng ZHOU School of Information and Engineering, Jingdezhen Ceramic University, Jiangxi 333403, P. R. China
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Let \{$X$, $X_n$, $n\ge 1$\} be a sequence of i.i.d. random vectors with ${\mathbb{E}}X=(0,\ldots,0)_{m\times 1}$ and ${\rm Cov}(X,X)=\sigma^2I_m$, and set $S_n=\sum_{i=1}^{n}X_i$, $n\ge 1$. For every $d>0$ and $a_n=o((\log\log n)^{-d})$, the article deals with the precise rates in the genenralized law of the iterated logarithm for a kind of weighted infinite series of ${\mathbb{P}}(|S_n|\ge (\varepsilon+a_n)\sigma \sqrt{n}(\log\log n)^d)$.